After entering my URL I could see, that Facebook somehow messed up the base path of the images. It turned out, that Facebook seems not to respect <base href="..." /> for the URLs of the page. As all URLs of the images on my page were given relative like <img src="typo3temp/yag/..." /> I got things like mimi.kaktusteam.de/path/to/my/blog/post/typo3temp/yag/... which hardly made any sense.

So the solution to this problem is simply adding an absolute path to your images: